Contributions to Seymour's Second Neighborhood Conjecture

TL;DR

This paper investigates Seymour's Second Neighborhood Conjecture by establishing conditions for the existence of vertices with non-decreasing second neighborhood size and exploring properties of minimal counterexamples.

Contribution

It provides new sufficient conditions for the conjecture and analyzes the structure of minimal graphs that do not satisfy it.

Findings

Existence of conditions ensuring some vertex v with |N_1(v)| ≤ |N_2(v)|

Minimal graphs without such vertices have specific structural properties

If such minimal graphs exist, infinitely many strongly-connected graphs lack such vertices

Abstract

Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there must exist some v in V(D) such that |N_1(v)| is less than or equal to |N_2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex.